Related papers: An existence time estimate for K\"ahler-Ricci flow

Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed…

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by…

We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci…

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on $\mathbb{R}^{n+1}$ that are noncollapsed at infinity, without assuming bounded…

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the…

Let $(M,J,\Omega)$ be a closed polarized complex manifold of K\"ahler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of K\"ahler metrics that are invariant under $G$ and represent the…

In this paper, we announce the following results: Let M be a Kaehler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler-Ricci…

We propose a construction of mean curvature flows by approximation for very general initial data, in the spirit of the works of Brakke and of Kim & Tonegawa based on the theory of varifolds. Given a general varifold, we construct by…

Let $X$ be a compact K\"ahler manifold. We prove that the K\"ahler-Ricci flow starting from arbitrary closed positive $(1,1)$-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has…

In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary,…

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This…

In this paper we survey the recent developments of the Ricci flows on complete noncompact K\"{a}hler manifolds and their applications in geometry.

In this paper we prove a local curvature estimate for the $\kappa$-LYZ flow over K\"ahler manifolds introduced in [FGP1] and [LYZ]. In particular, we generalize the long time existence of the flow.

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can…

We study the convergence of the K\"ahler-Ricci flow on a compact K\"ahler manifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futaki invariant on $\pi c_1(M;J)$. As the application we establish a criterion for the…

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow…

In this short paper, we show that K\"ahler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some mild assumption.

The main result of this paper is: Given any constant C, there is $(\epsilon,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(\epsilon,k,L)$-neck, then the…

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the…

We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on…